Vector current correlator to two loops in chiral perturbation theory.

Abstract

The isospin-breaking correlator of the product of flavor octet vector currents, ${\mathrm{\ensuremath{\Pi}}}_{\mathrm{\ensuremath{\mu}}\ensuremath{\nu}}^{38}$(${\mathit{q}}^{2}$)= i\ensuremath{\int}${\mathit{d}}^{4}$x exp(iq\ensuremath{\cdot}x)〈0\ensuremath{\Vert}T[${\mathit{V}}_{\mathrm{\ensuremath{\mu}}}^{3}$ (x)${\mathit{V}}_{\ensuremath{\nu}}^{8}$(0)]\ensuremath{\Vert}0〉, is computed to next-to-next-to-leading (two-loop) order in chiral perturbation theory. Large corrections to both the magnitude and ${\mathit{q}}^{2}$ dependence of the one-loop result are found, and the reasons for the slow convergence of the chiral series for the correlator given. The two-loop expression involves a single O(${\mathit{q}}^{6}$) counterterm, present also in the two-loop expressions for ${\mathrm{\ensuremath{\Pi}}}_{\mathrm{\ensuremath{\mu}}\ensuremath{\nu}}^{33}$(${\mathit{q}}^{2}$) and ${\mathrm{\ensuremath{\Pi}}}_{\mathrm{\ensuremath{\mu}}\ensuremath{\nu}}^{88}$(${\mathit{q}}^{2}$), which counterterm contributes a constant to the scalar correlator ${\mathrm{\ensuremath{\Pi}}}^{38}$(${\mathit{q}}^{2}$), defined by ${\mathrm{\ensuremath{\Pi}}}_{\mathrm{\ensuremath{\mu}}\ensuremath{\nu}}^{38}$(${\mathit{q}}^{2}$)\ensuremath{\equiv}(${\mathit{q}}_{\mathrm{\ensuremath{\mu}}}$${\mathit{q}}_{\ensuremath{\nu}}$-${\mathit{q}}^{2}$${\mathit{g}}_{\mathrm{\ensuremath{\mu}}\ensuremath{\nu}}$) ${\mathrm{\ensuremath{\Pi}}}^{38}$(${\mathit{q}}^{2}$). The feasibility of extracting the value of this counterterm from other sources is discussed. Analysis of the slope of the correlator with respect to ${\mathit{q}}^{2}$ using QCD sum rules is shown to suggest that, even to two-loop order, the chiral series for the correlator may not yet be well converged, and the physical origins of possible important O(${\mathit{q}}^{8}$) contributions discussed. \textcopyright{} 1996 The American Physical Society.